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Mirrors > Home > MPE Home > Th. List > gagrp | Structured version Visualization version GIF version |
Description: The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
gagrp | ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2733 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2733 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 1, 2, 3 | isga 19140 | . . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))) |
5 | 4 | simplbi 499 | . 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V)) |
6 | 5 | simpld 496 | 1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 × cxp 5670 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 +gcplusg 17184 0gc0g 17372 Grpcgrp 18806 GrpAct cga 19138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-map 8810 df-ga 19139 |
This theorem is referenced by: gafo 19145 gass 19150 galcan 19153 gacan 19154 gapm 19155 gaorber 19157 gastacl 19158 galactghm 19256 sylow2alem2 19470 |
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